Hypotheses

Along with the three main hypotheses sets (Maximizing, Traditioanl, and General), we have two additional sets. These sets are importantly different in that the entire parameter space is not defined. They are also defined a bit differently.

Threshold

This hypothesis set is based on defining each diel period in terms of lower thresholds. If the thresholds are not provided, default values are used.

triplot(hyp=hyp.sets("Threshold"))

Not that where there is no colored space indicates that this set of probabilities are not considered as possible parameter values. Therefore, when using this hypothesis set you are deciding that these combinations of parameter values can not occur.

Variation

This hypothesis set is based on defining the ranges (lower, upper) of possible probability values.

triplot(hyp=hyp.sets("Variation"))

User-modified Hypotheses

You do not need to rely on pre-set definitions of diel hypotheses. If you are interested in categorizing dominant diel phenotypes based on 0.90 probability or higher, you can adjust the hypotheses using the function ‘diel.ineq()’ than pass this object to the plot function or model fitting function. Here, we will just define the lower bount of each diurnal, nocturnal, and crepuscular using \(\xi_{1} = 0.90\)

  diel.setup=diel.ineq(xi = c(0.90))
#> Warning in diel.ineq(xi = c(0.9)): length(xi) == 1 when it should be 2.
#> Calculated the second value as (1 - xi)/2,
#> where xi is the single value you provided.
  triplot(hyp=hyp.sets("Traditional"),diel.setup=diel.setup)

If you want to separate the hypotheses and reduce the parameter space of cathemerality,

  diel.setup=diel.ineq(xi = 0.70,separation=0.10)
#> Warning in diel.ineq(xi = 0.7, separation = 0.1): length(xi) == 1 when it should be 2.
#> Calculated the second value as (1 - xi)/2,
#> where xi is the single value you provided.
  triplot(hyp=hyp.sets("Traditional"),diel.setup=diel.setup)

When considering the General hypothesis set, you can adjust the lower threshold probability value (\(\xi_{1}\)) to define crepuscular, diurnal, and nocturnal, as well as adjust the definitions of Cathemeral General vs Diurnal-Nocturnal, Crepuscular-Nocturnal, and Diurnal-Crepuscular (\(\xi_{2}\)). To do this, set \(xi\) as a vector of two probabilities,

  diel.setup=diel.ineq(xi = c(0.9,0.05))
  triplot(hyp=hyp.sets("General"),diel.setup=diel.setup)

This new hypothesis set can be used to simulate data and fit the same models by passing the object diel.setup as an attribute of the functions sim.diel and diel.fit.


y=sim.diel(hyp="D.CR",n.sample = 200,diel.setup=diel.setup)$y

out = diel.fit(y,hyp.set=hyp.sets("General"),n.chains=2,post.fit = TRUE,diel.setup=diel.setup)
#> Data checks Complete.
#> Calculating Bayes Factors...
#> Posterior Sampling...
#> The most supported model is: 
#>  Diurnal-Crepuscular (General)

triplot(out)

There are additional defaults that could be changed, depending on your hypotheses of interest.

#To see all the default inputs
  defaults=diel.ineq()
  defaults$inputs
#> $e.D
#> [1] 0.05
#> 
#> $e.N
#> [1] 0.05
#> 
#> $e.CR
#> [1] 0.05
#> 
#> $e.EC
#> [1] 0.05
#> 
#> $e.AV
#> [1] 0.1
#> 
#> $xi.t.D
#> [1] 0.8
#> 
#> $xi.t.N
#> [1] 0.8
#> 
#> $xi.t.CR
#> [1] 0.8
#> 
#> $xi.t.C
#> [1] 0.2
#> 
#> $eta.D
#> [1] 0.9
#> 
#> $eta.N
#> [1] 0.9
#> 
#> $eta.CR
#> [1] 0.9
#> 
#> $eta.C
#> [1] 0.333
#> 
#> $xi
#> [1] 0.8 0.1
#> 
#> $p.avail
#> [1] 0.1666660 0.4166667
#> 
#> $separation
#> [1] 0

For the Variation hypotheses, we can adjust the range of values for all hypotheses as,


# Default
  triplot(hyp =hyp.sets("Variation"))

# New triplot(hyp =hyp.sets("Variation"),diel.setup=diel.ineq(e = 0.2)) # You can change individual hypotheses as well. Here, we are changing the range of the diurnal hypothesis only. This is not recommend. But, there is flexibility to be creative. triplot(hyp =hyp.sets("Variation"),diel.setup=diel.ineq(e.D = 0.4))

For the Threshold hypotheses, we can adjust the lower probability similarly as,

  
# Default
  triplot(hyp =hyp.sets("Threshold"))

# Adjusted triplot(hyp =hyp.sets("Threshold"),diel.setup=diel.ineq(xi.t.D = 0.6, xi.t.N = 0.6, xi.t.CR = 0.6))

User-specified Hypotheses

Next, we will demonstrate how to include additional diel hypotheses. This requires understanding the inequality constraint setup and then coding the results as additions to the ‘diel.ineq’ output. Let’s consider two complementary hypotheses that account for the complete parameter space among the three fundamental probabilities. Our objective is to evaluate whether an animal is highly crepuscular or not. We will define highly crepuscular as \(p_{tw} \geq 0.5\). The complementing hypothesis is that the animal does not have such high crepuscular activity \(p_{tw} < 0.5\).

Hypothesis 1 Inequalities

\[ \begin{equation} \begin{split} & p_{\text{tw}} \geq 0.5 \\ \end{split} \end{equation} \]

Next, we need to translate each inequality into the framework of \(\mathbf{A} \boldsymbol{\theta} \leq \mathbf{b}\).

\[ \begin{equation} \begin{split} & p_{\text{tw}} \geq 0.50 \\ & -p_{\text{tw}} \leq -0.50 \\ & (-1) \times p_{\text{tw}} + (0)\times p_{\text{d}} \leq -0.50 \\ \end{split} \end{equation} \] Putting it together, we take the constants in parentheses left of the equal sign and package it into matrix \(\mathbf{A}\) and take the constants on the right of the equal sign and package it into vector \(\mathbf{b}\) as,

\[ \begin{equation} \textbf{A} = \begin{bmatrix} -1 & 0 \\ \end{bmatrix}, \textbf{b} = \begin{bmatrix} -0.50 \\ \end{bmatrix}. \end{equation} \]

Hypothesis 2 Inequalities

The complementing low activity during twilight as,

\[ \begin{equation} \begin{split} & p_{\text{tw}} \leq 0.49999 \\ \end{split} \end{equation} \] Following the same procedure, we find that,

\[ \begin{equation} \textbf{A} = \begin{bmatrix} 1 & 0 \\ \end{bmatrix}, \textbf{b} = \begin{bmatrix} 0.49999 \end{bmatrix}. \end{equation} \] ## Coding

Now, we need to code these hypotheses into \(\texttt{Diel.Niche}\)

# First, we need to create new list elements. One for each new hypothesis, with these ordered elements in each list,

# Element 1: Name 
# Element 2:  A
# Element 3:  b
# Element 4:  func

# Note, func - this should almost always be 'bf_multinom', unless an equality statement is specified, which required function 'bf_equality'

  A <- matrix(c(-1,0),ncol = 2, byrow = TRUE)
  b <- c(-0.5)
  New1=list(Name="Highly Crepuscular",A=A,b=b,func="bf_multinom")  

  A <- matrix(c(1,0),ncol = 2, byrow = TRUE)
  b <- c(0.49999)
  New2=list(Name="Not highly Crepuscular",A=A,b=b,func="bf_multinom")  

#Second, include these new lists into diel.ineq() function as,
  
  diel.setup=diel.ineq()
  diel.setup$New1=New1
  diel.setup$New2=New2

We have setup our hypotheses. Now, we need to visualize them to check it’s what we intended. Note that in each following, we pass our new diel.setup object to each function.

triplot(hyp=c("New1","New2"),diel.setup = diel.setup,more.points = TRUE)

Simulating and Fitting

Let’s simulate data under the hypothesis that the animal is highly crepuscular.

set.seed(43243)
dat=sim.diel(n.sample=85, hyp="New1",diel.setup = diel.setup)
y=dat$y
y
#>      y_crep y_day y_night
#> [1,]     61    22       2

Next, let’s compare our two models

#Note, that we get a warning, which is fine
out <- diel.fit(y, hyp.set = c("New1","New2"),diel.setup=diel.setup)
#> Data checks Complete.
#> Calculating Bayes Factors...
#> Warning in full.hyp.name(ms.model): Hypothesis code is not recognized
#> The most supported model is: 
#>  New1

#Let's look at the model probabilities, where we find the model we simulated is highly supported
out$bf.table
#>      Prior Posterior
#> New1   0.5         1
#> New2   0.5         0

Additional Hypotheses

We have included a few additional hypotheses that could be useful but don’t fit nicely into a hypothesis set that is only made up of mutually exclusive hypotheses. This generally means that larger sample sizes will be needed to identify them as most supported within a set.

Even Cathemarlity

A strict definition of cathemerality is that there is even amount of activity in all thre diel periods. Thus, the probabilities of activity of twilight, day, and night are all equal. We can consider this model by simulating data under this model and considering a combination of the Traditional hyptheses set and the even cathemarlity hypothesis(‘EC’).

set.seed(5452)
dat = sim.diel(n.sim=1, n.sample=100, hyp="EC")

y=dat$y

out = diel.fit(y, hyp.set=c(hyp.sets("Traditional"),"EC"), post.fit = TRUE)
#> Data checks Complete.
#> Calculating Bayes Factors...
#> Posterior Sampling...
#> The most supported model is: 
#>  Even Cathemeral Equality

out$bf.table
#>    Prior  Posterior
#> D    0.2 0.00000000
#> N    0.2 0.00000000
#> CR   0.2 0.00000000
#> C    0.2 0.02832742
#> EC   0.2 0.97167258

#Notice how small the parameter space is for the posteriors
plot(out)

Diel-Niche-Additional

Additional Package Details